Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

Triangle Midpoints

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

Fermat's Poser

Find the point whose sum of distances from the vertices (corners) of a given triangle is a minimum.

Age 14 to 16 Short Challenge Level:

The diagram on the right shows a square with side length $2\text{m}$ and lines drawn to its sides from its centre $O$. The points $A$, $B$, $C$ and $D$ are all on different sides of the square.

The lines $OA$ and $OB$ are perpendicular, as are $OC$ and $OD$.

If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.